Question

Five mole of a diatomic gas is kept at temperature\[T\]. The volume of the gas varies according to the law, \[V = a{T^{ - 2}}\], where a is positive constant. The final temperature of the gas is found to be \[5T\], what amount of heat is supplied to the gas?
A. \[5RT\]
B. \[(5/2)RT\]
C. \[(10/3)RT\]
D. \[10RT\]

Answer 1

Hint:As we know that for the diatomic gas, number of molecules are 5, initial temperature is T. Also V varies as \[V = a{T^{ - 2}}\] where $a$ is positive constant and also final temperature is 5T. The atomicity is the number of atoms combined to form the molecule to remain in the stable state. Mono-atomic gas are the atoms which have 1 atomicity and diatomic has 2 atomicity.

Complete step by step answer:
Number of mole\[ = 5mole\]
Temperature = $T$ (Initial temperature)
\[V = a{T^{ - 2}}\](Here V is the volume of gas)
Final temperature = 5T
As we know that
\[\dfrac{{PV}}{T} = {\text{constant = c}}\]
So, \[\dfrac{{PV}}{T} = {\text{constant = c}}\]
\[\dfrac{{PV}}{C} = T\]
Substitute the value of temperature, we get-
\[V = a{T^{ - 2}}\]
We can write-
\[V{T^2} = a\]
\[\Rightarrow P \times {\left( {\dfrac{{PV}}{C}} \right)^2} = a\]
\[\Rightarrow{P^2}{V^3} = {\text{constant}}\]
\[\Rightarrow P{V^{\dfrac{3}{2}}} = {C_2}\]----- (1)

Now compare this equation with \[P{V^N} = {\text{Constant}}\], we get-
\[N = \dfrac{3}{2}\]
Now for finding the value of C (Molar heat capacity)
\[C = ({C_V} - \dfrac{R}{{N - 1}})\]
\[\Rightarrow C = \dfrac{{5R}}{2} - \dfrac{R}{{\dfrac{3}{2} - 1}}\]
\[\Rightarrow C = \dfrac{R}{2}\]
Now, heat supplied,\[\Delta Q = n \times C \times \Delta T\]
Here $Q$ is the heat supplied, $N$ is the number of moles, $C$ is the heat capacity and ∆t is the change in initial and final temperature.

By substituting all the values, we get,
\[5 \times \dfrac{R}{2} \times \left( {5T - T} \right)\]
\[\Rightarrow 5 \times \dfrac{R}{2} \times \left( {4T} \right)\]
\[\therefore 10RT\]

So, option (D) is correct.

Note: Heat capacity is an extensive property. It is a physical property which is defined as the amount of heat that is supplied to the given mass of material to produce a unit change in its temperature. SI unit of heat capacity is \[Joules/k\].It is also known as thermal capacity.